20 
ference of the circle (1,1). The coefficient-function is then defined 
by the integral 
1 
zei foe LEAS an elven ORN) 
(1,1) 
taken round the just mentioned circle, and this integral is easily 
seen to be equal to the binomial series (1) in the half-plane of « 
on the right of the imaginary axis, provided definite agreements as 
to the value of 4! be made. It may, however, be difficult to 
investigate whether the integral (6) is equal to the given function 
w (az). And in any case the investigation is intricate if 2’ > — 1, 
especially when the difference between 2’ and —1 is rather great. 
For the coefficient-function of p(t) is then brought into relation with 
that of another generating function, with 2’ < — 1, by means of a 
polynomial consisting of a very large number of terms. 
The question therefore naturally suggests itself, if it is possible, 
to. find simpler tests which are sufficient for a function to be 
expanded in a binomial series. This is, indeed, the case, and we 
may, moreover, say that the obtained properties are about necessary *). 
For we can prove the following theorem: 
If a function w (w) is regular in the finite part of the half-plane 
R(#)>y¥?) (y= real), and if, in that domain, tt satisfies the inequality 
ONES ANC ot) oe ee 
where M is a positwe and l and 6 are real numbers, the latter such 
that b+ y>0, further a a complex number on the circumference 
of the circle (1,1), variable with the argument w of «—y, the argu- 
ment a of a being equal to —w so that 
WS DCO TE ed EE eat) 
then, in the domain (8 = real > v). 
R(@) Loe.) ae Ae 
1) For the sake of comparison we observe that for the expansion of a function 
in a series of factorials 
i n! an 
ETEN Dele in) 
there is a necessary and sufficient condition, stated by NirrseN and simplified by 
Pincuerte, which has some similarity with the above condition for binomial series. 
But it is not possible to find simple tests for the expansion of a function in a 
series of factorials. The only simple sufficient condition which may be given in 
this case, is that a function can be expanded in a series of factorials, if it is 
regular and of zero value at infinity. But this condition is far from necessary. 
2?) R(x) means the real part of a. 
